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Properties

Label 2-63-63.16-c1-0-1
Degree 22
Conductor 6363
Sign 0.996+0.0815i0.996 + 0.0815i
Analytic cond. 0.5030570.503057
Root an. cond. 0.7092650.709265
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.335 − 0.580i)2-s + (−0.377 + 1.69i)3-s + (0.775 − 1.34i)4-s + 1.42·5-s + (1.10 − 0.347i)6-s + (2.21 + 1.44i)7-s − 2.38·8-s + (−2.71 − 1.27i)9-s + (−0.477 − 0.827i)10-s − 4.93·11-s + (1.97 + 1.81i)12-s + (−1.37 − 2.38i)13-s + (0.0972 − 1.77i)14-s + (−0.537 + 2.40i)15-s + (−0.752 − 1.30i)16-s + (0.559 + 0.969i)17-s + ⋯
L(s)  = 1  + (−0.236 − 0.410i)2-s + (−0.217 + 0.975i)3-s + (0.387 − 0.671i)4-s + 0.637·5-s + (0.452 − 0.141i)6-s + (0.837 + 0.546i)7-s − 0.841·8-s + (−0.905 − 0.425i)9-s + (−0.151 − 0.261i)10-s − 1.48·11-s + (0.570 + 0.524i)12-s + (−0.381 − 0.661i)13-s + (0.0259 − 0.473i)14-s + (−0.138 + 0.621i)15-s + (−0.188 − 0.326i)16-s + (0.135 + 0.235i)17-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((0.996+0.0815i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0815i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+1/2) L(s)=((0.996+0.0815i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0815i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: 0.996+0.0815i0.996 + 0.0815i
Analytic conductor: 0.5030570.503057
Root analytic conductor: 0.7092650.709265
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ63(16,⋅)\chi_{63} (16, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :1/2), 0.996+0.0815i)(2,\ 63,\ (\ :1/2),\ 0.996 + 0.0815i)

Particular Values

L(1)L(1) ≈\approx 0.845532−0.0345271i0.845532 - 0.0345271i
L(12)L(\frac12) ≈\approx 0.845532−0.0345271i0.845532 - 0.0345271i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(0.377−1.69i)T 1 + (0.377 - 1.69i)T
7 1+(−2.21−1.44i)T 1 + (-2.21 - 1.44i)T
good2 1+(0.335+0.580i)T+(−1+1.73i)T2 1 + (0.335 + 0.580i)T + (-1 + 1.73i)T^{2}
5 1−1.42T+5T2 1 - 1.42T + 5T^{2}
11 1+4.93T+11T2 1 + 4.93T + 11T^{2}
13 1+(1.37+2.38i)T+(−6.5+11.2i)T2 1 + (1.37 + 2.38i)T + (-6.5 + 11.2i)T^{2}
17 1+(−0.559−0.969i)T+(−8.5+14.7i)T2 1 + (-0.559 - 0.969i)T + (-8.5 + 14.7i)T^{2}
19 1+(2.00−3.47i)T+(−9.5−16.4i)T2 1 + (2.00 - 3.47i)T + (-9.5 - 16.4i)T^{2}
23 1−5.43T+23T2 1 - 5.43T + 23T^{2}
29 1+(−3.40+5.89i)T+(−14.5−25.1i)T2 1 + (-3.40 + 5.89i)T + (-14.5 - 25.1i)T^{2}
31 1+(1.25−2.17i)T+(−15.5−26.8i)T2 1 + (1.25 - 2.17i)T + (-15.5 - 26.8i)T^{2}
37 1+(−0.709+1.22i)T+(−18.5−32.0i)T2 1 + (-0.709 + 1.22i)T + (-18.5 - 32.0i)T^{2}
41 1+(−0.124−0.215i)T+(−20.5+35.5i)T2 1 + (-0.124 - 0.215i)T + (-20.5 + 35.5i)T^{2}
43 1+(0.498−0.863i)T+(−21.5−37.2i)T2 1 + (0.498 - 0.863i)T + (-21.5 - 37.2i)T^{2}
47 1+(−4.73−8.20i)T+(−23.5+40.7i)T2 1 + (-4.73 - 8.20i)T + (-23.5 + 40.7i)T^{2}
53 1+(0.410+0.710i)T+(−26.5+45.8i)T2 1 + (0.410 + 0.710i)T + (-26.5 + 45.8i)T^{2}
59 1+(−3.29+5.70i)T+(−29.5−51.0i)T2 1 + (-3.29 + 5.70i)T + (-29.5 - 51.0i)T^{2}
61 1+(0.0376+0.0651i)T+(−30.5+52.8i)T2 1 + (0.0376 + 0.0651i)T + (-30.5 + 52.8i)T^{2}
67 1+(−6.29+10.9i)T+(−33.5−58.0i)T2 1 + (-6.29 + 10.9i)T + (-33.5 - 58.0i)T^{2}
71 1−0.0804T+71T2 1 - 0.0804T + 71T^{2}
73 1+(−5.34−9.25i)T+(−36.5+63.2i)T2 1 + (-5.34 - 9.25i)T + (-36.5 + 63.2i)T^{2}
79 1+(−0.922−1.59i)T+(−39.5+68.4i)T2 1 + (-0.922 - 1.59i)T + (-39.5 + 68.4i)T^{2}
83 1+(7.23−12.5i)T+(−41.5−71.8i)T2 1 + (7.23 - 12.5i)T + (-41.5 - 71.8i)T^{2}
89 1+(−6.76+11.7i)T+(−44.5−77.0i)T2 1 + (-6.76 + 11.7i)T + (-44.5 - 77.0i)T^{2}
97 1+(−2.70+4.67i)T+(−48.5−84.0i)T2 1 + (-2.70 + 4.67i)T + (-48.5 - 84.0i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−15.15220572324695818656001059970, −14.17379468297214502677178870637, −12.50901304462440725403087225598, −11.18016019411116836439550005621, −10.43512906466667548914009454451, −9.595433739461642443887281324057, −8.179445350993106075371674004931, −5.88249907646017099821222604099, −5.09911585727316482119125511722, −2.55410182050563309062851569300, 2.43276306555582225040876355560, 5.23491954184880197744602844310, 6.83534364245412976352260609106, 7.64492662535632723445952402770, 8.759029338414712789645250969721, 10.67025941076567510576401190927, 11.69984246674261101499035914297, 12.92015519761970628177184036033, 13.69176548065970242328307141540, 14.98446795700252637283799713248

Graph of the ZZ-function along the critical line